Strange duality on ${\mathbb{P}}^2$ via quiver representations

Abstract

We study Le Potier's strange duality conjecture on ${\mathbb{P}}^2$. We focus on the strange duality map $S{D}_{c_n^r,d}$ which involves the moduli space of rank $r$ sheaves with trivial first Chern class and second Chern class $n$, and the moduli space of 1-dimensional sheaves with determinant ${\mathcal{O}}_{{\mathbb{P}}^2}(d)$ and Euler characteristic 0. By using tools in quiver representation theory, we show that $S{D}_{c_n^r,d}$ is an isomorphisms for $r=n$ or $r=n-1$ or $d\le 3$, and in general $S{D}_{c_n^r,d}$ is injective for any $n\ge r>0$ and $d>0$.